In this paper, we present the Residual Integral Solver Network (RISN), a novel neural network architecture designed to solve a wide range of integral and integro-differential equations, including one-dimensional, multi-dimensional, ordinary and partial integro-differential, systems, fractional types, and Helmholtz-type integral equations involving oscillatory kernels. RISN integrates residual connections with high-accuracy numerical methods such as Gaussian quadrature and fractional derivative operational matrices, enabling it to achieve higher accuracy and stability than traditional Physics-Informed Neural Networks (PINN). The residual connections help mitigate vanishing gradient issues, allowing RISN to handle deeper networks and more complex kernels, particularly in multi-dimensional problems. Through extensive experiments, we demonstrate that RISN consistently outperforms not only classical PINNs but also advanced variants such as Auxiliary PINN (A-PINN) and Self-Adaptive PINN (SA-PINN), achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. These results highlight RISN's robustness and efficiency in solving challenging integral and integro-differential problems, making it a valuable tool for real-world applications where traditional methods often struggle.

翻译：本文提出了一种新颖的神经网络架构——残差积分求解网络（RISN），用于求解包括一维、多维、常积分-微分、偏积分-微分、方程组、分数阶类型以及涉及振荡核的Helmholtz型积分方程在内的广泛积分与积分-微分方程。RISN将残差连接与高斯求积、分数阶导数运算矩阵等高精度数值方法相结合，使其能够获得比传统物理信息神经网络（PINN）更高的精度和稳定性。残差连接有助于缓解梯度消失问题，使RISN能够处理更深层的网络和更复杂的核函数，尤其适用于多维问题。通过大量实验，我们证明RISN不仅持续优于经典PINN，也超越了辅助PINN（A-PINN）和自适应PINN（SA-PINN）等先进变体，在各类方程上均实现了显著更低的平均绝对误差（MAE）。这些结果凸显了RISN在求解具有挑战性的积分与积分-微分问题时的鲁棒性和高效性，使其成为传统方法往往难以应对的实际应用场景中的有力工具。